Table of zeros of the modified Bessel function of the second kind

bessel_zeros.md

K. V. Leung and S. S. Ghaderpanah (1979) An Application of the Finite Element Approximation Method to Find the Complex Zeros of the Modified Bessel Function Kn(z). Math. Comp. 33 (148), p. 1302 tabulates all zeros of the principal value of Kn⁡(z), for n=2⁢(1)⁢10 (not checked for OCR errors):

n           R(z)                            ± I(z)
2 -1.2813737976560964788484347943  4.2948496520871968779577284110
3 -1.6817888047458454647221805246  1.3080120322739490396502284389
4 -1.9781618634659070241700412324  2.2043719815468711802301895462
4 -2.6286711679571242189239356978  4.3269664862177846518524769588
5 -2.2186262746398760466991398972  3.1130829449859484932024599078
5 -3.1351328447046434221402224695  1.3038823977137057333868902412
6 -2.4234043880011252372669320326  4.0309615812693082518160866112
6 -3.5510979040000786787689256194  2.1834951775778858208942180135
6 -3.9615580702543404860650878072  4.3334540861473783228197493649
7 -2.6031262658681676073605916898  4.9559696065385237377946547292
7 -3.9081257398031834786414725236  3.0708717702488955667126782799
7 -4.5126267774997091289243634516  1.3027788416202445155997188405
8 -2.7641429773113422214747067049  5.8867128822557099755500071045
8 -4.2231522789550338555580428982  3.9650659693874916655430042023
8 -4.9882787925531101790986431543  2.1770827464789907589102679543
8 -5.2907612925994479853857895222  4.3357769522400748475850727130
9 -2.9105824231361973641127863752  6.8221903314329512927258712281
9 -4.5064659455202975444947754224  4.8652071432637217649541918277
9 -5.4097474475400236316635820253  3.0565442393434597827549120928
9 -5.8665514666851772591442131577  1.3023283269981807534615329324
10 -3.0452934989589489096726573840 7.7616556708745682219592817911
10 -4.7648453733729039927675961572 5.7705555987099768475527429057
10 -5.7900271641796772769369101588 3.9409726157692468366207406587
10 -6.3783949707941569023306522990 2.1742485862026654802191073802
10 -6.6184818847079360692456212008 4.3368620578620986460836833209

However, they omitted the exponents (2nd order imaginary part should be 4.29...e-1 = 0.429, for instance), and only about 18 of the 29 digits shown for each zero are accurate. mpmath can take the existing values and improve their accuracy:

mp.dps = 30
for N in approx_zeros:
    print(N)
    for approx_zero in approx_zeros[N]:
        zero = mp.findroot(lambda x: mp.besselk(N, x), approx_zero)
        print(zero)

2
(-1.28137379765609647606842843702 + 0.429484965208719699979811867829j)
3
(-1.68178880474584545852964452107 + 1.30801203227394905226347115023j)
4
(-1.97816186346590701567141374268 + 2.20437198154687119327978222537j)
(-2.62867116795712421753133102013 + 0.43269664862177847742327657008j)
5
(-2.21862627463987603641106715533 + 3.11308294498594850666931389254j)
(-3.13513284470464341867149427782 + 1.30388239771370574576401824707j)
6
(-2.4234043880011252254959067352 + 4.03096158126930826567724829612j)
(-3.55109790400007867373547881176 + 2.18349517757788583344571093194j)
(-3.96155807025434048513628270226 + 0.433345408614737844529913837749j)
7
(-2.60312626586816759431219413351 + 4.95596960653852375202809648794j)
(-3.90812573980318347233376332359 + 3.07087177024889557944997968152j)
(-4.51262677749970912649088292552 + 1.30277884162024452791648739715j)
8
(-2.76414297731134220729762156005 + 5.88671288225570999013570598961j)
(-4.22315227895503384816535595624 + 3.96506596938749167846876998834j)
(-4.98827879255311017545859453395 + 2.17708274647899077132326501838j)
(-5.29076129259944798468907403114 + 0.433577695224007497009906317208j)
9
(-2.91058242313619734892019308636 + 6.82219033143295130764590223583j)
(-4.50646594552029753615130376854 + 4.86520714326372177806727658457j)
(-5.40974744754002362700723842337 + 3.05654423934345979527718736024j)
(-5.86655146668517725726526229157 + 1.30232832699818076575399968447j)
10
(-3.04529349895894889355396408651 + 7.7616556708745682371976385773j)
(-4.76484537337290398357402725511 + 5.77055559870997686085023387688j)
(-5.79002716417967727139695659505 + 3.94097261576924684925875324259j)
(-6.37839497079415689946518591002 + 2.17424858620266549257300556782j)
(-6.61848188470793606868820651115 + 0.433686205786209876861367361584j)

For example:

• original value:
  • besselk(10, -6.6184818847079360692456212008 + 4.3368620578620986460836833209e-1i)
  • Result is ≈ 10⁻¹⁷
• Improved value:
  • besselk(10, -6.61848188470793606868820651115 + 0.433686205786209876861367361584*i)
  • Result is ≈ 10⁻³⁰